Kinematics and Dynamics of Rotational Motion

Q: What is the difference between angular velocity and linear velocity?

Angular velocity \omega measures how fast the body's angle changes (rad/s), and is the same for every particle. Linear velocity v = r\omega measures how fast a specific point moves along its circular path (m/s), and depends on the point's distance r from the axis. Points on the axis have v = 0; points on the rim have the highest v.

Q: Why does a particle in circular motion at constant angular speed still have acceleration?

Because the direction of its velocity changes continuously, even though the magnitude is constant. This direction change is an acceleration - the centripetal acceleration a_c = v^2/r = r\omega^2, directed toward the centre of the circle. Only when both magnitude and direction of \vec{v} are constant is acceleration zero.

Q: What is the difference between tangential and centripetal acceleration?

Tangential acceleration a_t = r\alpha is along the direction of motion and changes the speed of the particle. Centripetal acceleration a_c = r\omega^2 is perpendicular to velocity (pointing inward) and changes only the direction. Both exist simultaneously when a particle in circular motion is speeding up or slowing down.

Q: When can I use the constant angular acceleration equations?

Only when \alpha is genuinely constant over the interval of interest. Examples include a wheel spinning down under constant friction, or a motor providing constant torque on a body of fixed moment of inertia. If \alpha depends on time (e.g. torque varies), integrate the definitions directly instead of using the kinematic equations.

Q: How do I convert rpm to rad/s?

\omega \text{(rad/s)} = 2\pi \times N / 60, where N is in rpm. For example, 300 rpm equals 2\pi \times 300 / 60 = 10\pi \approx 31.4 rad/s. Always convert to rad/s before applying v = r\omega or the kinematic equations.

Q: Is angular velocity a vector?

Yes, in the following sense: \vec{\omega} is directed along the axis of rotation, with sense given by the right-hand rule (curl the fingers along the rotation, thumb points along \vec{\omega}). Infinitesimal angular displacements d\vec{\theta} add as vectors, so \vec{\omega} is a genuine vector. But finite angular displacements do not commute, so they are not vectors.

PhysicsSystem Of Particles And Rotational MotionFor JEE aspirants

Rotational motion describes how a rigid body turns about a fixed axis. The key angular variables are angular displacement $θ$ , angular velocity $ω = d θ / d t$ , and angular acceleration $α = d ω / d t$ . These play the same role as $x, v, a$ in linear motion, and satisfy the same-shape kinematic equations when $α$ is constant: $ω = ω_{0} + α t$ , $θ = ω_{0} t + \frac{1}{2} α t^{2}$ , $ω^{2} = ω_{0}^{2} + 2 α θ$ . Every particle of a rigid body has the same $ω$ and $α$ at a given instant. The relations $v = r ω$ and $a_{t} = r α$ connect the angular quantities to the linear speed and tangential acceleration of a particle at distance $r$ from the axis. These fundamentals are essential for JEE Main, JEE Advanced, and NEET rotational mechanics.

Key Formulas - Quick Reference

Angular velocity: $ω = \frac{d θ}{d t}$ ; SI unit rad/s
Angular acceleration: $α = \frac{d ω}{d t} = \frac{d ^{2} θ}{d t ^{2}}$ ; SI unit rad/s²
Kinematic equations (constant $α$ ): $ω = ω_{0} + α t$
$θ = ω_{0} t + \frac{1}{2} α t^{2}$
$ω^{2} = ω_{0}^{2} + 2 α θ$
Arc length: $s = r θ$
Tangential (linear) speed: $v = r ω$
Tangential acceleration: $a_{t} = r α$ ; Centripetal: $a_{c} = r ω^{2} = v^{2} / r$
$ω$ in rad/s from rpm: $ω = \frac{2 π N}{60}$

1. Rigid Body and Types of Motion

A rigid body is an idealised object in which the distance between any two points remains constant with time - shape and size do not change however the body moves. A brick, a wooden bat, and a spinning top are excellent rigid bodies; a jelly, a stretched spring, or a stream of water are not.

A rigid body can undergo three kinds of motion:

Pure translation: every particle of the body has the same velocity and acceleration at every instant. A box sliding on a level floor is in pure translation.
Pure rotation about a fixed axis: every particle moves in a circle whose centre lies on the axis; all circles have the same axis. A ceiling fan is in pure rotation.
Combined (rolling) motion: the axis of rotation itself moves. A rolling wheel translates and rotates.

In this concept we focus on the first two, in particular the kinematics of rotation about a fixed axis and the dynamical equations governing rotational motion about that axis. Rolling and combined motion are treated in later concepts.

2. Angular Displacement

Consider a rigid body rotating about an axis perpendicular to the page, passing through a point $O$ . Suppose two arbitrary points $A$ and $B$ of the body move to positions $A^{'}$ and $B^{'}$ after some time. Because the body is rigid, $O A = O A^{'}$ and $O B = O B^{'}$ , and the angles turned by every particle about the axis are identical:

∠ A O A^{'} = ∠ B O B^{'} = θ

This common angle is called the angular displacement of the body. It is measured in radians. Because every particle undergoes the same angular displacement in the same time interval, we can describe the rotation of the entire body by a single variable $θ (t)$ .

Vector nature: Finite angular displacements do not add like vectors (rotations do not commute), but infinitesimal ones

d θ

do. Consequently

ω

and

α

are proper vectors, directed along the axis of rotation by the right-hand rule.

3. Angular Velocity

The average angular velocity over a time interval $[t_{1}, t_{2}]$ is

\overset{ω}{ˉ} = \frac{θ _{2} - θ _{1}}{t _{2} - t _{1}}

The instantaneous angular velocity is the limit as the interval shrinks:

ω = Δ t \to 0 lim \frac{Δ θ}{Δ t} = \frac{d θ}{d t}

SI unit is rad/s. Common non-SI unit: revolutions per minute (rpm), with $ω (rad/s) = 2 π N /60$ where $N$ is in rpm.

As a vector, $ω$ is directed along the axis of rotation. Curl the fingers of the right hand in the direction of rotation; the thumb points along $ω$ .

4. Angular Acceleration

The rate of change of angular velocity is the angular acceleration:

α = \frac{d ω}{d t} = \frac{d ^{2} θ}{d t ^{2}}

SI unit is rad/s². As a vector, $α$ is along $ω$ if the body is speeding up and opposite to $ω$ if it is slowing down (about a fixed axis).

5. Equations of Rotational Kinematics

When $α$ is constant, integrating gives equations of exactly the same form as linear kinematics, with the substitutions $x \to θ$ , $v \to ω$ , $a \to α$ :

Linear motion	Rotational motion
$v = u + a t$	$ω = ω_{0} + α t$
$x = u t + \frac{1}{2} a t^{2}$	$θ = ω_{0} t + \frac{1}{2} α t^{2}$
$v^{2} = u^{2} + 2 a x$	$ω^{2} = ω_{0}^{2} + 2 α θ$
$x = \frac{1}{2} (u + v) t$	$θ = \frac{1}{2} (ω_{0} + ω) t$

These are valid only if $α$ is constant over the time interval. If $α$ varies with time, integrate the definitions directly.

Solved Example 1

A disc starts rotating from rest with a constant angular acceleration

α = π

rad/s² about a fixed axis perpendicular to its plane through its centre. Find (a) the angular velocity after 4 s, (b) the angular displacement in 4 s, (c) the number of revolutions completed in 4 s.

Solution:

Given $ω_{0} = 0$ , $α = π$ rad/s², $t = 4$ s.

(a) $ω = ω_{0} + α t = 0 + π (4) = 4 π$ rad/s.

(b) $θ = ω_{0} t + \frac{1}{2} α t^{2} = 0 + \frac{1}{2} π (16) = 8 π$ rad.

(c) Number of revolutions $n = θ / (2 π) = 8 π / (2 π) = 4$ revolutions.

6. Relation Between Linear and Angular Variables

Consider a particle $A$ of a rigid body at distance $r$ from the axis of rotation. As the body turns through an angle $θ$ , the particle traces an arc of length

s = r θ

Figure 1: A particle at distance

r

from the axis. Linear speed

v = ω r

(tangential), and centripetal acceleration

a_{c} = ω^{2} r

(radially inward).

Differentiating with respect to time:

Tangential speed: $v = r ω$ (perpendicular to the radius, along the direction of motion).
Tangential acceleration: $a_{t} = r α$ (along the tangent, in the direction of motion if speeding up).
Centripetal (radial) acceleration: $a_{c} = r ω^{2} = v^{2} / r$ (radially inward, exists whenever the particle moves in a circle, even at constant $ω$ ).

The total linear acceleration of the particle is the vector sum: $a = a_{t} + a_{c}$ , with magnitude $a = a_{t}^{2} + a_{c}^{2}$ .

Same $ω$ , different $v$ : Every particle of a rigid body rotating about a fixed axis shares the same

ω

and

α

at any instant. But the linear speed

v = r ω

varies from zero (on the axis) to maximum (at the outermost radius). This is why the tip of a fan blade moves much faster than points near the hub, even though the whole blade rotates as one.

Solved Example 2

A wheel of radius 0.5 m starts from rest and reaches an angular speed of 20 rad/s in 4 seconds. Find (a) the angular acceleration, (b) the linear speed of a point on the rim at

t = 4

s, (c) the tangential and centripetal accelerations of that point at

t = 4

Solution:

(a) $α = (ω - ω_{0}) / t = (20 - 0) /4 = 5$ rad/s².

(b) $v = r ω = 0.5 \times 20 = 10$ m/s.

(c) $a_{t} = r α = 0.5 \times 5 = 2.5$ m/s². $a_{c} = r ω^{2} = 0.5 \times 400 = 200$ m/s². These are perpendicular, so the total linear acceleration magnitude is $2. 5^{2} + 20 0^{2} \approx 200$ m/s².

7. Vector Form of Linear-Angular Relations

For a particle at position vector $r$ measured from any point on the axis, the vector form of the linear-angular relations is

v = ω \times r, a_{t} = α \times r, a_{c} = ω \times v = ω \times (ω \times r) = - ω^{2} r_{⊥}

where $r_{⊥}$ is the component of $r$ perpendicular to $ω$ (i.e. the radius of the circular path). The right-hand rule for the cross product gives the correct direction of $v$ automatically.

8. Introduction to Torque

To change the rotational state of a rigid body about a fixed axis, we need an angular equivalent of force. That quantity is torque $τ$ , defined as

τ = r \times F

where $r$ is the position vector of the point of application of force $F$ from the axis (or a point on the axis). The magnitude is

τ = r F sin θ = F \cdot d

where $θ$ is the angle between $r$ and $F$ , and $d = r sin θ$ is the perpendicular distance from the axis to the line of action of the force (called the moment arm). SI unit is N m (dimensionally the same as joule but never expressed in joules; torque is a different physical quantity from work).

What torque does: Torque is to rotation what force is to translation. Just as

F

produces linear acceleration

a

, torque

τ

produces angular acceleration

α

through the rotational Newton's second law

τ = I α

, where

I

is the moment of inertia. This law is developed in detail in a later concept.

9. Setting Up Rotational Problems

A typical problem in rotational kinematics or dynamics involves the following steps:

Identify the axis of rotation and check whether it is fixed.
List all initial conditions: $ω_{0}$ , $θ_{0}$ , and any given constraints.
Identify whether $α$ is constant (then use the kinematic equations directly) or varies with time (then integrate $α = d ω / d t$ ).
Use $v = r ω$ , $a_{t} = r α$ , $a_{c} = r ω^{2}$ to relate angular quantities to linear quantities of specific particles when needed.
For dynamics problems, apply $τ_{n e t} = I α$ about the axis (this uses the moment of inertia concept developed separately).

Solved Example 3

A grinding wheel initially rotating at 300 rpm is switched off. It comes to rest after making 50 complete revolutions. Assume the angular deceleration is constant. Find (a) the angular deceleration, and (b) the time taken to come to rest.

Solution:

Convert: $ω_{0} = 2 π (300) /60 = 10 π$ rad/s. Final $ω = 0$ . Angular displacement $θ = 50 \times 2 π = 100 π$ rad.

(a) Using $ω^{2} = ω_{0}^{2} + 2 α θ$ : $0 = (10 π)^{2} + 2 α (100 π)$ , so $α = - 100 π^{2} / (200 π) = - π /2$ rad/s². The magnitude of angular deceleration is $π /2$ rad/s².

(b) Using $ω = ω_{0} + α t$ : $0 = 10 π + (- π /2) t$ , giving $t = 20$ s.

Solved Example 4

A particle at distance

R

from the axis of a rotating body has linear speed

v_{1}

at time

t_{1}

and

v_{2}

at time

t_{2}

. If angular acceleration is constant, find its value.

Solution:

Angular velocities: $ω_{1} = v_{1} / R$ , $ω_{2} = v_{2} / R$ .

α = \frac{ω _{2} - ω _{1}}{t _{2} - t _{1}} = \frac{v _{2} - v _{1}}{R ( t _{2} - t _{1} )}

Common Mistakes to Avoid

Watch out

Confusing angular displacement in revolutions with radians. The kinematic equations require radians. Always convert: 1 rev = $2 π$ rad.
Applying the constant- $α$ kinematic equations ( $ω = ω_{0} + α t$ etc.) when $α$ is time-varying. If $α$ is not constant, integrate the definitions directly.
Forgetting that every particle of a rigid body has the same $ω$ but different linear speeds $v = r ω$ . Points on the axis are momentarily at rest even though the body rotates.
Using $v = r ω$ or $a_{t} = r α$ in RPM or degree units directly. These relations require $ω$ in rad/s and $α$ in rad/s².
Ignoring centripetal acceleration when the particle moves in a circle at constant $ω$ . Even at constant angular speed, a particle has $a_{c} = r ω^{2}$ directed toward the axis.
Treating finite angular displacements as vectors. Only infinitesimal angular displacements $d θ$ (and hence $ω$ and $α$ ) are true vectors; finite rotations do not commute.

Frequently Asked Questions

Q1. Why is angular velocity the same for every particle of a rigid body?

Because the body is rigid, any two particles maintain a fixed separation and orientation. Any turn of the body carries every particle through the same angle in the same time interval. So $ω = d θ / d t$ is a property of the body, not of a particular particle.

Q2. What is the difference between angular velocity and linear velocity?

Angular velocity $ω$ measures how fast the body's angle changes (rad/s), and is the same for every particle. Linear velocity $v = r ω$ measures how fast a specific point moves along its circular path (m/s), and depends on the point's distance $r$ from the axis. Points on the axis have $v = 0$ ; points on the rim have the highest $v$ .

Q3. Why does a particle in circular motion at constant angular speed still have acceleration?

Because the direction of its velocity changes continuously, even though the magnitude is constant. This direction change is an acceleration - the centripetal acceleration $a_{c} = v^{2} / r = r ω^{2}$ , directed toward the centre of the circle. Only when both magnitude and direction of $v$ are constant is acceleration zero.

Q4. What is the difference between tangential and centripetal acceleration?

Tangential acceleration $a_{t} = r α$ is along the direction of motion and changes the speed of the particle. Centripetal acceleration $a_{c} = r ω^{2}$ is perpendicular to velocity (pointing inward) and changes only the direction. Both exist simultaneously when a particle in circular motion is speeding up or slowing down.

Q5. When can I use the constant angular acceleration equations?

Only when $α$ is genuinely constant over the interval of interest. Examples include a wheel spinning down under constant friction, or a motor providing constant torque on a body of fixed moment of inertia. If $α$ depends on time (e.g. torque varies), integrate the definitions directly instead of using the kinematic equations.

Q6. How do I convert rpm to rad/s?

$ω (rad/s) = 2 π \times N /60$ , where $N$ is in rpm. For example, 300 rpm equals $2 π \times 300/60 = 10 π \approx 31.4$ rad/s. Always convert to rad/s before applying $v = r ω$ or the kinematic equations.

Q7. Is angular velocity a vector?

Yes, in the following sense: $ω$ is directed along the axis of rotation, with sense given by the right-hand rule (curl the fingers along the rotation, thumb points along $ω$ ). Infinitesimal angular displacements $d θ$ add as vectors, so $ω$ is a genuine vector. But finite angular displacements do not commute, so they are not vectors.

Previous year questions on Kinematics and Dynamics of Rotational Motion

9 questions from past papers, each with a step-by-step solution.

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